Remark 4 The FunPrinSS probability is also connected to the subspace sampling of [1]}. To perform the CUR decomposition for a matrix \(\mathbf {X}\in \mathbb {R}^{N\times p}\) , they compute the row sampling probabilities as follows. A matrix \(\mathbf {C}\in \mathbb {R}^{N\times q}\)  is formed by sampling columns from \(\mathbf {X}\) , and its SVD \(\mathbf {C}= \mathbf {U}_C\mathbf {D}_C\mathbf {V}_C^T\)  is computed. Then, the sampling probability for \(n\) -th row of \(\mathbf {X}\)  is computed from the \(n\) -th row of \(\mathbf {U}_C\) , the \(n\) -th row of \(\mathbf {X}^{\perp }_C = (\mathbf {I}-\mathbf {U}_C\mathbf {U}_C^T)\mathbf {X}\)  and their product
\(p_n = \frac{(1/3)\Vert (\mathbf {U}_C)_{(n)}\Vert _2^2}{\sum _{m = 1}^N \Vert (\mathbf {U}_C)_{(m)}\Vert _2^2} +\frac{(1/3)\Vert (\mathbf {U}_C)_{(n)}\Vert _2\Vert (\mathbf {X}^{\perp }_C)_{(n)}\Vert _2}{\sum _{m = 1}^N \Vert (\mathbf {U}_C)_{(m)}\Vert _2\Vert (\mathbf {X}^{\perp }_C)_{(m)}\Vert _2}+\frac{(1/3)\Vert (\mathbf {X}^{\perp }_C)_{(n)}\Vert _2^2}{\sum _{m = 1}^N \Vert (\mathbf {X}^{\perp }_C)_{(m)}\Vert _2^2}.\) 
